Kimball Tutorial 1 Tutorial 2 ⇒

Content: Using the exact solution of the Schrödinger equation for the ground state of the H atom, derive the Ansatz of G.E. Kimball ES 17 December 2018//2002//1982

Properties of the H1s wavefunction

H atom, 1s ground state (spatial) wavefunction: Proton at origin, r distance of electron from proton

Exact H1s (spatial) wavefunction in atomic units! e is the base of the natural logarithms.

H atom ground state electron probability density

H atom ground state probability density summed over all space. Must be 1, meaning: We will find the electron of the atom with certainty somewhere in the universe

The same two functions in 3D:

This shows the electron density cusp at the location of the proton

Plot of the radial density of the H atom ground state: This is the point density at r multiplied with the volume of an infinitesimal spherical shell dr, plotted as function of r. Dimension wise, it is the charge in that spherical shell at r. Its maximum is at r = 1 a0, the radius of the 1s orbit of the atom model of Niels Bohr. The difference of Bohr’s model to the quantum chemical model is the curve below against one very high stick at exactly 1 [r/a0] (a Dirac delta function!). Furthermore, in Bohr’s model the electron has an angular momentum relative to an axis through the barycenter of the atom, which has not been observed in the 1s ground state.

The next image is the surface of revolution of the above curve around the z-axis

with proton at the origin

H atom, average distance of electron from proton R, bohr units

Inside this root mean-square distance we find about 58% of the total charge of 1 electron:

and at 5.7 times that distance = 450 pm the atom is nearly complete, hence atoms, while formally infinite, are in practice minute:

Energy components:

H atom, potential energy

H atom, total potential energy

H atom, kinetic energy

H atom, total kinetic energy

H atom, total Energy

H atom, ground state equilibrium energy terms expressed with R, the expectation value of the distance of the electron from the proton

(You can do the following few rows of arithmetic easily on paper by pencil!)

Kimball’s “Ansatz”: Total energy Etot = T+V

T has dimension [/m]* = [m**], dim[V] is [*], hence:

and should be a minimum! Set its derivative to zero to find the optimal R=x:

Substitute x for R:

We have already determined, that Etot = -1/2 Eh, and R=3/2 a0.

Solve for the constants a,b using these results:

Substitute a,b into the “Ansatz” above and obtain Kimball’s equation for the H atom in its stationary ground state:

This is exact! It is a significant mathematical transformation of Schrödinger’s equation for the H-atom ground state, contains the same information and has been misunderstood by all practitioners of a qualitative Kimball and Tangent Sphere models:

Kinetic energy Eh: Full dimension [/m]*

Potential energy Eh:

Virial Theorem

Check the ratio

This should be -2.00... by the Virial Theorem, which is valid in classical and quantum physics. The value -2 of this ratio signifies, that the force maintaining the observed system in a state of equilibrium varies proportional to . It is the electrostatic force between every charged particle pair of the system! If computations of molecules and solid lattices are reasonably complete, they yield Vir of 2.000±0.002. Most texts on quantum chemistry give a proof of this theorem.

Note: In this tutorial I have used the qualifier “exact” for the H1s (spatial) wavefunction and the “Kimball equation”. Those two expressions are equivalent. They are “exact” in the usual context of quantum chemical computations, but not “absolutely true”. We have neglected the finite kinetic energy of the proton (see Tutorial2 and 6), which is ~1/1836 of the kinetic energy of the electron, and have assumed that the proton is a point charge with a size negligible compared to the “size” of the H-atom. Furthermore, all movements within the atom leave the barycenter fixed. We have neglected relativistic effects, quantum electrodynamic corrections, magnetic interactions between electron and proton spins, and others. They are all of an absolute size or less of the values shown.

Finally, this H1s (spatial) wavefunction is the (real valued) amplitude function which goes through zero once during every period of the “wave”, i.e. the particle vanishes and reappears again - addio conservation of mass! This catastrophe is eliminated by the clever mathematics of the complete 1s wavefunction which is complex-valued with a complex time factor. It is often told, that the “square” of a wavefunction φ at a certain location q (of configuration space) is the probability for finding the particle whose state is φ(q) at q. However, this pertains to the real part of the product φ*(q)φ(q) of two complex conjugate factors. This prevents the particle described by the wavefunction φ(q) to change between death and birth during every period! I do not know of any textbook with “wave mechanics for the high school” which describes this correctly! It would help to tell this and thus avoid the erroneous notion that quantum mechanical “waves” have macroscopic analogues. Hence, “wave mechanics for the high school” is a fake: The pupils (perhaps with exceptions, like young Wolfgang Pauli) just do not have the mathematical skill to understand Hermitian operators and work with them.- You may find a careful discussion of these topics in the first 37 pages of P.W. Atkins, Molecular Quantum Mechanics, 2nd ed, Oxford, 1983.